What's actually happening on the level of individual particles when an atom in a solid is vibrating? For context, I'm trying to understand what happens to the kinetic energy of an electron in semiconductor after it absorbs a photon with more energy than the bandgap.  My QM professor said it gets converted to thermal energy, i.e. atomic vibrations, but is being frustratedly vague about HOW this happens.
As I was thinking about it, I realized I've never considered how atomic vibrations work w.r.t. QM at all.  I'm assuming they don't literally vibrate but that that's just a oversimplification. It makes sense of we think of atoms as single solid objects, like in the crystal lattice model of a semiconductor, but of of course atoms aren't actually like that -- they're mostly empty space (at least, ignoring the issue of vacuum fluctuations in QFT) and the particles they're composed of are best modeled by wavefunctions.  I can't picture how something like that could "vibrate" in the usual sense of the word.
Also, if kinetic energy from an individual electron is converted to thermal energy of the crystal lattice, as my professor says, that seems to imply it's somehow transferred to the nucleus (since it has to have transferred somewhere, and most of the mass of an atom is from the nucleus, so if there's vibration of any sort happening, I'd think it'd have to be mainly happening with the nucleus), but I don't see how that's possible -- don't the quantized energy levels mean the electrons in an atom can't directly interact with the nucleus?  If so, how could energy from an electron be transfer to the nucleus?
 A: Qualitatively:
Atoms  are quantum mechanical entities. They are described by a wavefunction  which gives the probability in space time for the atom to exist at (x,y,z,t). An electron hitting a single atom will interact quantum mechanically with the whole atom , the Feynman diagram being with the fringe field of the electrons it has. Depending on the energy of the electron, it could be an elastic scatter, the electron could transfer some energy etc. All interactions can be modeled.
In a similar way, the lattice occupied by many atoms has to be modeled quantum mechanically, because in principle, one wavefunction describes the lattice quantum mechanically and all its interactions, the electron in your case, are quantum mechanical. Due to the huge number of atoms in a lattice, quantum mechanical models have developed, as the band theory of solids. This gives the behavior in terms of energy for the electrons in the solid.

One of the implications of the symmetric lattice of atoms is that it can support resonant lattice vibration modes. These vibrations transport energy and are important in the thermal conductivity of non-metals, and in the heat capacity of all solids.

That is where phonons come in as far as your question is concerned.
You say:

Also, if kinetic energy from an individual electron is converted to thermal energy of the crystal lattice, as my professor says, that seems to imply it's somehow transferred to the nucleus (since it has to have transferred somewhere, and most of the mass of an atom is from the nucleus, so if there's vibration of any sort happening,

You are thinking in classical terms of a problem that has only quantum mechanical modeling, it is the space location of the whole atom that will be vibrating as far as the quantum mechanical lattice goes.
A: First things first
At $T=0$, the ion cores (i.e. the nucleus of the atom + electrons in lower shells) form a perfect lattice (assuming no impurities). As Bloch Theorem shows, electrons (or better: Bloch waves) do not scatter in a perfect lattice. So even though both the electron and the ion cores have non-zero net charge, they do not interact. This is very important!
How atoms can vibrate
This is a bit of a weird question for me. I know quantum mechanics is strange and can be unintuitive, but don't toss your intuition out of the window. Think back to how one goes about solving the Schrödinger equation for a single atom. The first approximation one makes is the Born-Oppenheimer approximation. It uses the fact that the nucleus is much heavier than the electron shell. So in most cases, one can think of the position of the atom as a whole in a classical picture: vibration or oscillation of an atom is described by $\langle\mathbf{R}\rangle = A\cos\omega t$, where $\mathbf{R}$ is the position operator of the center of mass of the atom.
Phonons (classical and quantum mechanics)
To be precise, the word phonons as a quasi-particle only makes sense within quantum mechanics. But, in order to get there, let's start with a classical treatment. The ion cores arange into a lattice, because there is repulsive interaction between them (they have the same charge). In the simplest case, the repulsive energy is minimized when they align in a perfect cubic lattice (also called simple cubic, sc). To lowest order, the potential a single ion core feels is harmonic. When it is brought out of equilibrium, it oscillates around that potential minimum; a classical simple pendulum. In a lattice, this movement also excites the other ion cores to oscillate. Due to periodic boundary conditions, only a discrete set of motional frequencies is allowed. As we always do, we want to find the eigenmodes and eigenenergies of the system (still classical treatement here!). This is were optical and acoustic phonons come from. In this classical picture, the word phonon refers to these eigenmodes of oscillation. To turn this into quantum mechanics, we can treat the displacement of the ionic cores as a field and quantize it canonically. The eigenmodes are delocalized over the whole crystal. The way to think about electron-phonon interaction is to imagine a localized electron and phonon (superposition of many Bloch waves and phonons). One can picture this localized phonon as a positive charge cloud the negatively charged electron interacts with. Now we have turned electron-phonon interaction into a scattering problem of a negative and positive charged particle.
The important thing to take home is that electrons DO NOT INTERACT WITH THE LATTICE (edit: more precise is "they do not scatter from the lattice). You can't think of it as scattering of an electron with a single atom. You have to think in this delocalized phonon picture, which disrupt the perfect lattice structure and serve as scattering sources. Electrons only interact with lattice imperfections They allow the excited electron to relax into a lower energy state. If there were no phonons (i.e. $T=0$) or lattice defects, the electron could only relax by emission of a photon (or electron-electron scattering, but that still leaves you with an electron in an excited state).
